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A102574 a(n) is the sum of the distinct norms of the divisors of n over the Gaussian integers.

%I A102574 #36 Feb 13 2024 02:19:26
%S A102574 1,7,10,31,31,70,50,127,91,217,122,310,183,350,310,511,307,637,362,
%T A102574 961,500,854,530,1270,781,1281,820,1550,871,2170,962,2047,1220,2149,
%U A102574 1550,2821,1407,2534,1830,3937,1723,3500,1850,3782,2821,3710,2210,5110,2451
%N A102574 a(n) is the sum of the distinct norms of the divisors of n over the Gaussian integers.
%C A102574 Also sum of divisors of n^2 which are the sum of two squares (A001481). For example the divisors of 3^2 are 1, 3, 9 of which only 1 and 9 are in A001481 and a(3) = 1 + 9 = 10. - _Jianing Song_, Aug 03 2018
%H A102574 Amiram Eldar, <a href="/A102574/b102574.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)
%H A102574 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gaussian_integer">Gaussian integer</a>.
%F A102574 a(n) = sigma_2(A097706(n)) * sigma((n/A097706(n))^2). - _Andrew Howroyd_, Aug 03 2018
%F A102574 Multiplicative with a(p^e) = sigma(p^(2e)) = (p^(2e+1) - 1)/(p - 1) if p = 2 or p == 1 (mod 4); sigma_2(p^e) = (p^(2e+2) - 1)/(p^2 - 1) if p == 3 (mod 4). - _Jianing Song_, Aug 03 2018
%F A102574 Sum_{k=1..n} a(k) ~ c * n^3, where c = (5/12) * zeta(3) * A243380 = 0.52812367275583317729... . - _Amiram Eldar_, Feb 13 2024
%e A102574 Let ||i|| denote the norm of i.
%e A102574 a(2) = 1 + ||1+i|| + 2^2 = 1 + 2 + 4 = 7.
%e A102574 a(5) = 1 + ||1+2i|| + 5^2 = 1 + 5 + 25 = 31. Note that ||1+2i|| = ||2+i|| so their norm (5) is only counted once.
%t A102574 b[n_] := Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}];
%t A102574 a[n_] := With[{r = b[n]}, DivisorSigma[2, r] DivisorSigma[1, (n/r)^2]];
%t A102574 a /@ Range[50] (* _Jean-François Alcover_, Sep 20 2019, from PARI *)
%o A102574 (PARI) \\ here b(n) is A097706.
%o A102574 b(n)={my(f=factor(n)); my(r=prod(i=1, #f~, my([p,e]=f[i,]); if(p%4==3, p^e, 1))); r}
%o A102574 a(n)={my(r=b(n)); sigma(r,2)*sigma((n/r)^2)} \\ _Andrew Howroyd_, Aug 03 2018
%o A102574 (Python)
%o A102574 from math import prod
%o A102574 from sympy import factorint
%o A102574 def A102574(n): return prod((q := int(p & 3 == 3))*(p**(2*(e+1))-1)//(p**2-1) + (1-q)*(p**(2*e+1)-1)//(p-1) for p, e in factorint(n).items()) # _Chai Wah Wu_, Jun 28 2022
%Y A102574 Cf. A000203 (sigma), A001157 (sigma_2), A001481, A097706, A103230, A243380.
%K A102574 nonn,mult,easy
%O A102574 1,2
%A A102574 _Yasutoshi Kohmoto_, Feb 25 2005
%E A102574 Corrected and extended by _David Wasserman_, Apr 08 2008
%E A102574 Keyword:mult added by _Andrew Howroyd_, Aug 03 2018
%E A102574 Name clarified by _Jianing Song_, Aug 03 2018